PSC HSST Statistics Model Questions and Answers

76. A 95% confidence interval for λ, when a large sample is taken from a Poisson population with parameter λ is
(A) `bar x “+- 1.65“sqrt(( bar x)/(n)) `
(B) `lambda+- 1.65 sqrt((lambda)/(n)) `
(C) `bar x+- 1.96sqrt((bar x)/(n)) `
(D) `lambda +- 1.96sqrt((lambda)/(n)) `
Answer: C

77. The minimum Chi-squared estimators are not necessarily
(A) Unbiased
(B) Consistent
(C) Efficient
(D) Asymptotically normal
Answer: A

78. Which one of the following statements is true?
(A) Even if the UMP test does not exist, a UMPU test may exist
(B) Even if the UMPU test does not exist, a UMP test may exist
(C) A UMP test exists only if a UMPU test exists
(D) A UMPU test exists only if a UMP test exists
Answer: A

79. In paired `t` test the two random variables should be
(A) Paired and uncorrelated
(B) Unpaired and correlated
(C) Both paired and correlated
(D) Neither paired nor correlated
Answer: C

80. With usual notations, the criterion for acceptance in SPRT is
(A) `lambda_m <= ((1-beta))/(alpha) `
(B) `lambda_m >= ((1-beta))/(alpha) `
(C) `lambda_m <=(beta)/((1-alpha)) `
(D) ` lambda_m >= (beta)/((1-alpha)`
Answer: C

81. The Poisson process with parameter λ is a renewal counting process for which the unit lifetimes have ……………. distribution with common parameter λ.
(A) Poisson
(B) Exponential
(C) Uniform
(D) Geometric
Answer: B

82. Let `{X_n, n = 0, 1, 2…}` be a Branching process and the corresponding offspring distribution has a pgf `P (s)=(2)/(3) +(s+s^2)/(6) ` . Find the probability of extinction of the process
(A) 0
(B) 0.25
(C) 0.66
(D) 1
Answer: D

83. Let `{X_n}` be a renewal process with `mu = E (X_1) <oo ` and if `M (t)` is the renewal function, then `lim_(t->oo) (M (t))/(t) = …..`
(A) `(1)/(mu) `
(B) `mu `
(C) `(t)/(mu) `
(D) `(mu)/(t) `
Answer: A

84. If `X_i`’s are independent Poisson variates with respective parameters `lambda_i`, for `i = 1, 2, …k`, then the conditional distribution of `X_1, X_2,… X_k` given their sum `sum_(i=1)^k X_i = n` is a …………….. distribution with parameters ………………….. and ………………….
(A) Binomial with parameters `n` and ` (1)/(k)`
(B) Binomial with parameters `k` and `(1)/(n)`
(C) Multinomial with parameters `n` and `(1)/(k)`
(D) Multinomial with parameters `k` and `(1)/(n)`
Answer: C

85. If ` (X_1, X_2)` is a Bivariate normal random vector with parameters `(mu_{X1}, mu_{X2}, sigma ^2_X_1,sigma^2_X_2, rho`), when `sigma^2_X_1 = sigma^2_X_2 ` and `rho = 0` , the density function is called
(A) Elliptical Normal
(B) Circular Normal
(C) Symmetrical Normal
(D) Uniform Normal
Answer: B

86. If the random vector `X` follows Multivariate Normal distribution with mean vector 0 and dispersion matrix `I` and `Q_i = X^’ A_i X` are quadratic forms of rank `r_i` such that `sum_(i=1)^k A_i = I_p` , then a necessary and sufficient condition for `Q_i`’s to be distributed as independent chi-square random variables with `r_i` d.f is that
(A) ` sum_(i=1)^k r_i=k`
(B) `sum_(i=1)^k r_i = p`
(C) `sum_(i=1)^k r_i=0`
(D) `sum_(i=1)^k r_i = kp`
Answer: B

87. The relationship between partial correlation coefficients `r_{ij.k},` multiple correlation
coefficients `R_{i.jk}`and simple correlation coefficients `r_{ij}` is
(A) `R^2_1.23 = 1+ (1-r^2_12) (1 – r^2_13.2)`
(B) `R^2_1.23 = 1 – (1- r^2_12) (1- r^2_13.2)`
(C) `R^2_1.23 = 1 + (1-r^2_12)// (1-r^2_13.2)`
(D) `R^2_1.23 = 1- (1-r^2_12) // (1-r^2_13.2)`
Answer: B

88. Hotelling’s `T^2` statistic and Mahalnobis `D^2` statistic are connected by the relationship
(A) `D^2 = ((N_1 N_2))/((N_1+N_2)) T^2`
(B) `D^2 =((N_1 N_2))/((N_1-N_2)) T^2`
(C) `D^2 = ((N_1-N_2))/((N_1 N_2)) T^2`
(D) `D^2 = ((N_1 + N_2))/((N_1 N_2)) T^2`
Answer: D

89. In principal component analysis the variances of the Principal Components are the ……………….. of the covariance matrix.
(A) diagonal elements
(B) eigen values
(C) normalized elements
(D) non-zero elements
Answer: B

90. For discriminating between two populations R.A. Fisher suggested the linear discriminant function `X’l` for which
(A) ` (“(mean difference)”^2)/(“variance”)`
(B) `(“mean difference”)^2/(“A.M.”)`
(C) `(“mean difference”)/(“median”)`
(D) `(“variance”)/(“mean difference”)`
Answer: A

91. Assume that the time to failure `(T)` for a certain bulb has an exponential distribution `f ((t)/(lambda))` with parameter `lambda >0` with the prior pdf `g (lambda)` of `lambda` is an exponential distribution with parameter 2. Then the posterior pdf of `lambda` given `T = t` is
(A) `(2)/(t+2)`
(B) `(lambda)/(e^lambda (t+2))`
(C) `(lambda e^(lambda (t+2)))/((t+2)^2)`
(D) `(lambda (t+2)^2)/(e^(lambda (t+2)))`
Answer: D

92. The basic elements of statistical decision theory is
(A) a space Ω `= {ul theta}` of all possible states of nature
(B) an action space `A = {a}` of all possible courses of action
(C) a loss function `L (ul theta, a)` giving the incurred loss when action `a` is taken and the state is ` ul theta`
(D) all these
Answer: D

93. When there is no censoring for the life length `T`, the general formula of a survival function is
(A) `hat {S (t)} = (” # of individuals with” T >= t)/(“total sample size”)`
(B) `hat {S (t)} = (” # of individuals with” T <= t)/(“total sample size”)`
(C) `hat {S (t)} = (” # of individuals with” T = t)/(“total sample size”)`
(D) `hat {S (t)} = (” # of individuals with” T = 0)/(“total sample size”)`
Answer: A

94. The Cox’s Proportional Hazard Model (Cox’s PH Model) with explanatory variables ` ul X = (X_1, X_2, … X_p),beta_i` their regression coefficients and `h_0 (t)` a base line hazard, is `h (t, ul X) =`
(A) `e^{h_0 (t) sum_(i=1)^p beta_i X_i}`
(B) `log h_0 (t) + sum_(i=1)^p beta_i X_i`
(C) `h_0 (t) e sum_(i=1)^p beta_i X_i`
(D) `e^(h_0 (t)) sum_(i=1)^p beta_i X_i`
Answer: C

95. When an inspection lot contains no defectives the OC function `L (p)` is
(A) `L (p) = 1`
(B) `L (p) = oo`
(C) ` L (p) = 0`
(D) None of these
Answer: A

96. In a Time series data, the two main components which cause lack of stationarity are
(A) Seasonal and irregular variations
(B) Cyclic and irregular variations
(C) Trend and cyclic variations
(D) Trend and seasonal variations
Answer: D

97. In the ARMA (1, 1) model `Z_t = “phi Z_{t-1} + epsilon _t – theta epsilon_{t-1}` the condition for stationarity and invertilibility are respectively
(A) ` | phi | <= 1 and | theta | < 1` with `phi != theta`
(B) ` | phi | <= 1 and | theta | < 1` with `phi = theta`
(C) ` | phi | > 1 and | theta | > 1` with `phi!= theta`
(D) ` | phi | > 1 and | theta | > 1` with `phi = theta`
Answer: A

98. In a Linear programming Problem with `n + m` variables and `m` constraints the number of basic solutions is
(A) ` ((n+m),(m))`
(B) `((n),(m))`
(C) `((m),(n))`
(D) `((n+m),(n-m))`
Answer: A

99. If the demand curve is of the form `p = ae^{-bx}` , where `p` is the price and `x` is the demand, then the price elasticity of demand is
(A) `eta_p = bx`
(B) `eta_p = – bx`
(C) `eta_p = 1//bx`
(D) `eta_p = – 1 // bx`
Answer: C

100. The Engel’s curves for constant prices and those for constant incomes are respectively
(A) Concave and Convex
(B) Convex and Concave
(C) Both Concave
(D) Both Convex
Answer: B